A Generalized Forward-Backward Splitting

Abstract

Abstract. This paper introduces a generalized forward-backward splitting algorithm for finding a zero of a sum of maximal monotone operators B + n i=1 A i , where B is cocoercive. It involves the computation of B in an explicit (forward) step and of the parallel computation of the resolvents of the A i 's in a subsequent implicit (backward) step. We prove its convergence in infinite dimension, and robustness to summable errors on the computed operators in the explicit and implicit steps. In particular, this allows efficient minimization of the sum of convex functions f + n i=1 g i , where f has a Lipschitz-continuous gradient and each g i is simple in the sense that its proximity operator is easy to compute. The resulting method makes use of the regularity of f in the forward step, and the proximity operators of the g i 's are applied in parallel in the backward step. While the forwardbackward algorithm cannot deal with more than n = 1 non-smooth function, we generalize it to the case of arbitrary n. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.

A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

“…We note that another recent method has been proposed in [23] to solve (45), in the more restrictive setting where G = 0 and L i = I , for every i = 1, . .…”

“…In return, this quest of practitioners for efficient minimization methods has caused a renewed interest among mathematicians around splitting methods in monotone and nonexpansive operator theory, as can be judged from the numerous recent contributions, e.g. [13][14][15][16][17][18][19][20][21][22][23][24]. The most classical operator splitting methods to minimize the sum of two convex functions are the forward-backward method, proposed in [2] and further developed in [3,4,7,[25][26][27][28], and the Douglas-Rachford method [3,6,7,22].…”

Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

“…Since this algorithm is able to minimize a sum of three non-smooth convex functions, the constraint can be handled as a convex indicator function. Notice that other algorithms could be used such as the Generalized Forward Backward algorithm (Raguet et al 2013). …”

Total-activation regularized deconvolution of resting-state fMRI leads to reproducible networks with spatial overlap

“…Specifically, the spatial regularization promotes coherent activation within the regions of a predefined structural atlas while enforcing sparsity on the amount of active regions. We used the generalized forward-backward splitting algorithm in order to solve the regularization problem in temporal and space domains [22]. The algorithm was presented in detail in our previous work [20].…”